Subgroups — #26122107

Order 12032.

814 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689147	C₂	1	2	yes	maximal
2	C₂×#15688888	C₂	1	4	yes	maximal
3	C₂²×C₄×C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×#10625329	C₂²	1	4	yes	—
5	#15689147	—	2	4	no	—
6	#15689147	C₄	1	8	yes	—
7	C₂×C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
8	#15688888	—	2	8	no	—
9	C₂³×C₈×C₄₇	C₂²	1	2	yes	—
10	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₄₇	—	2	4	no	—
12	C₂×C₄×C₈×C₄₇	C₄	1	2	yes	—
13	C₂×C₄×C₈×C₄₇	C₄	1	2	yes	—
14	C₂×#10625329	—	2	4	no	—
15	C₂×#10625329	—	2	8	no	—
16	C₂×#10625329	—	2	8	no	—
17	C₂×#10625329	C₂×C₄	1	8	yes	—
18	C₂²×C₄₇⋊C₈	—	2	4	no	—
19	C₄⋊(C₄₇⋊C₈)	—	2	16	no	—
20	C₂³×C₄×C₄₇	Q₈	1	2	yes	—
21	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
22	C₂²×C₈×C₄₇	—	2	4	no	—
23	C₂²×C₈×C₄₇	—	2	4	no	—
24	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
25	C₂²×C₈×C₄₇	D₄	1	4	yes	—
26	C₂²×C₈×C₄₇	Q₈	1	4	yes	—
27	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
28	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₄₇	D₄	1	2	yes	—
31	C₂×C₄²×C₄₇	D₄	1	4	yes	—
32	C₄×C₈×C₄₇	—	2	4	no	—
33	C₄×C₈×C₄₇	—	2	4	no	—
34	C₂⁴×C₄₇	C₂×Q₈	1	1	yes	characteristic
35	C₂²×C₄×C₄₇	—	2	8	no	—
36	C₂²×C₄×C₄₇	C₄⋊C₄	1	8	yes	—
37	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
38	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
39	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
41	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
42	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
44	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
45	C₂²×C₄×C₄₇	#41	1	4	yes	—
46	C₂×C₈×C₄₇	—	2	4	no	—
47	C₂×C₈×C₄₇	—	2	4	no	—
48	C₂×C₈×C₄₇	—	2	8	no	—
49	C₂×C₈×C₄₇	—	2	8	no	—
50	C₂×C₈×C₄₇	C₄⋊C₄	1	8	yes	—
51	C₄²×C₄₇	C₄²	1	2	yes	—
52	C₄²×C₄₇	C₄²	1	2	yes	—
53	C₄²×C₄₇	C₂²⋊C₄	1	8	yes	—
54	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
55	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
56	#10625329	—	2	16	no	—
57	#10625329	—	4	8	no	—
58	C₂×C₄₇⋊C₈	—	2	8	no	—
59	C₂×C₄₇⋊C₈	—	4	4	no	—
60	C₂×C₄₇⋊C₈	—	4	4	no	—
61	C₂³×C₄₇	—	2	4	no	—
62	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
63	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂³×C₄₇	#125	1	2	yes	—
65	C₂³×C₄₇	C₄⋊Q₈	1	2	yes	—
66	C₂×C₄×C₄₇	—	2	4	no	—
67	C₂×C₄×C₄₇	—	2	16	no	—
68	C₂×C₄×C₄₇	#95	1	4	yes	—
69	C₂×C₄×C₄₇	#95	1	4	yes	—
70	C₂×C₄×C₄₇	#108	1	8	yes	—
71	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
72	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
73	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
75	C₂×C₄×C₄₇	#117	1	2	yes	—
76	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
77	C₂×C₄×C₄₇	C₄×Q₈	1	4	yes	—
78	C₂×C₄×C₄₇	#121	1	4	yes	—
79	C₂×C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
80	C₂×C₄×C₄₇	#124	1	2	yes	—
81	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
82	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
83	C₈×C₄₇	—	2	8	no	—
84	C₄₇⋊C₈	—	4	8	no	—
85	#10844	—	47	1	no	maximal, Sylow
86	C₂²×C₄₇	—	2	4	no	—
87	C₂²×C₄₇	—	2	4	no	—
88	C₂²×C₄₇	—	2	4	no	—
89	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
90	C₂²×C₄₇	#382	1	1	yes	characteristic
91	C₂²×C₄₇	#384	1	1	yes	characteristic
92	C₂²×C₄₇	#389	1	4	yes	—
93	C₂²×C₄₇	C₂×#108	1	4	yes	—
94	C₄×C₄₇	—	2	4	no	—
95	C₄×C₄₇	#341	1	8	yes	complemented
96	C₄×C₄₇	C₄×C₄⋊C₄	1	2	yes	—
97	C₄×C₄₇	#380	1	2	yes	—
98	C₄×C₄₇	#390	1	2	yes	—
99	C₄×C₄₇	#391	1	2	yes	—
100	C₂×#423	—	47	2	no	—
101	C₂²×C₄×C₈	D₄₇	1	1	yes	characteristic
102	C₂×#341	—	47	4	no	—
103	C₂×C₄₇	—	2	4	no	—
104	C₂×C₄₇	C₄⋊#95	1	1	yes	characteristic
105	C₂×C₄₇	C₂×#341	1	2	yes	—
106	C₂×C₄₇	#1837	1	2	yes	—
107	C₂×C₄₇	#1848	1	2	yes	—
108	C₂²×M₁₆	—	47	4	no	—
109	C₂³×C₈	C₂×D₄₇	1	2	yes	—
110	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
111	#423	—	47	8	no	—
112	#423	—	94	4	no	—
113	C₂×C₄⋊C₈	—	94	4	no	—
114	C₂×C₄×C₈	—	2	4	no	—
115	C₂×C₄×C₈	Dic₄₇	1	2	yes	—
116	C₂×C₄×C₈	Dic₄₇	1	2	yes	—
117	#341	—	94	8	no	—
118	C₄₇	#10844	1	1	yes	characteristic, complemented, Sylow
119	C₂³×C₄	Dic₉₄	1	2	yes	—
120	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
121	C₂×M₁₆	—	47	8	no	—
122	C₂×M₁₆	—	94	4	no	—
123	C₂×M₁₆	—	94	8	no	—
124	C₂×M₁₆	—	94	8	no	—
125	C₂²×C₈	—	2	4	no	—
126	C₂²×C₈	—	2	4	no	—
127	C₂²×C₈	—	94	4	no	—
128	C₂²×C₈	Dic₉₄	1	4	yes	—
129	C₂²×C₈	D₁₈₈	1	4	yes	—
130	C₂²×C₈	C₂×Dic₄₇	1	4	yes	—
131	C₂×C₄²	C₄×D₄₇	1	4	yes	—
132	C₂×C₄²	D₁₈₈	1	2	yes	—
133	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
134	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
135	C₂×C₄²	C₂²⋊D₄₇	1	4	yes	—
136	C₄⋊C₈	—	94	16	no	—
137	C₄×C₈	—	2	4	no	—
138	C₄×C₈	—	2	4	no	—
139	C₂⁴	C₂×Dic₉₄	1	1	yes	characteristic
140	C₂²×C₄	—	2	8	no	—
141	C₂²×C₄	Dic₄₇⋊C₄	1	8	yes	—
142	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
143	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
144	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
145	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
146	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
147	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
148	C₂²×C₄	#10625350	1	4	yes	—
149	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
150	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
151	M₁₆	—	94	16	no	—
152	M₁₆	—	188	8	no	—
153	C₂×C₈	—	2	4	no	—
154	C₂×C₈	—	2	4	no	—
155	C₂×C₈	—	2	8	no	—
156	C₂×C₈	—	2	8	no	—
157	C₂×C₈	—	94	8	no	—
158	C₂×C₈	—	188	4	no	—
159	C₂×C₈	—	188	4	no	—
160	C₂×C₈	C₄⋊Dic₄₇	1	8	yes	—
161	C₄²	C₄×Dic₄₇	1	2	yes	—
162	C₄²	C₄×Dic₄₇	1	2	yes	—
163	C₄²	(C₂×D₄₇)⋊C₄	1	8	yes	—
164	C₄²	C₂²⋊Dic₄₇	1	2	yes	—
165	C₄²	C₂²⋊Dic₄₇	1	2	yes	—
166	C₂³	—	2	4	no	—
167	C₂³	C₂×Dic₄₇⋊C₄	1	2	yes	—
168	C₂³	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
169	C₂³	#13094016	1	2	yes	—
170	C₂³	#13094024	1	2	yes	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	2	16	no	—
173	C₂×C₄	C₄₇⋊#95	1	4	yes	—
174	C₂×C₄	C₄₇⋊#95	1	4	yes	—
175	C₂×C₄	#13093960	1	8	yes	—
176	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
177	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	2	yes	—
178	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
179	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
180	C₂×C₄	#13093983	1	2	yes	—
181	C₂×C₄	C₄×D₁₈₈	1	4	yes	—
182	C₂×C₄	C₄×Dic₉₄	1	4	yes	—
183	C₂×C₄	#13093998	1	4	yes	—
184	C₂×C₄	#13094003	1	4	yes	—
185	C₂×C₄	#13094012	1	2	yes	—
186	C₂×C₄	#13094021	1	1	yes	characteristic
187	C₂×C₄	#13094024	1	1	yes	characteristic
188	C₈	—	2	8	no	—
189	C₈	—	188	8	no	—
190	C₂²	—	2	4	no	—
191	C₂²	—	2	4	no	—
192	C₂²	—	2	4	no	—
193	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
194	C₂²	#15688989	1	1	yes	characteristic
195	C₂²	#15688993	1	1	yes	characteristic
196	C₂²	#15689021	1	4	yes	—
197	C₂²	C₂×#13093960	1	4	yes	—
198	C₄	—	2	4	no	—
199	C₄	#15688888	1	8	yes	complemented
200	C₄	C₄×C₄⋊Dic₄₇	1	2	yes	—
201	C₄	#15688983	1	2	yes	—
202	C₄	#15689025	1	2	yes	—
203	C₄	#15689029	1	2	yes	—
204	C₂	—	2	4	no	—
205	C₂	#20241618	1	1	yes	characteristic
206	C₂	C₂×#15688888	1	2	yes	—
207	C₂	#20243526	1	2	yes	—
208	C₂	#20243560	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂²×C₄×C₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂×C₄⋊C₄
prime radical	C₂⁴×C₄₇	C₂×Q₈
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂²×C₄×C₈	D₄₇
hypocenter	C₄₇	#10844
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
95. C₄×C₄₇	#341	semidirect
118. C₄₇	#10844	semidirect
199. C₄	#15688888	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#15689147	2
C₂×#15688888	2
C₂²×C₄×C₈×C₄₇	2
#10844	47